Preheat your oven to 350 degrees F. Line a baking sheet with parchment paper and set aside.
In a medium mixing bowl, stir together the flour and salt. Set aside.
In a large mixing bowl, beat together the butter, sugar, and brown sugar until fluffy, about 1 minute. Add the egg and vanilla and mix to combine. Add the dry ingredients and mix until just combined.
Use a cookie scoop or measuring spoon to portion 1 tablespoonful scoops of cookie dough. Then use your hand to roll them into balls. Using the back of a teaspoon measuring spoon, make an indentation in the middle of each circle. (This will be the cup for the frosting.)
Bake for 12 minutes until the edges are also browned. Remove the cookies from the oven and cool them completely on a wire rack.
Frosting
In a medium mixing bowl, beat together the melted butter, powdered sugar, and rum until smooth.
Transfer the frosting to a piping bag or a food storage bag with one of the bottom corners cut off. Pipe a dollop of frosting into the indentation of each cookie. Sprinkle the frosting with a little ground nutmeg for garnish.
Notes
If you only have salted butter on hand, skip adding any salt to the cookie dough. You can swap in salted butter for the frosting with no issue.